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Chapter 6: Problem 37

$$ y=3 x-2 $$

Short Answer

Expert verified
The graph of \(y=3x-2\) is a straight line with a slope of 3 and y-intercept of -2. It passes through the points (0, -2) and increases by 3 on the y-axis for every increase of 1 on the x-axis.

Step by step solution

01

Identify the slope and y-intercept

In the equation \(y=3x-2\), the coefficient of \(x\) is the slope, which is 3. This means for every increase in 1 in \(x\), \(y\) will increase by 3. The y-intercept is the constant term, which is -2. The y-intercept is where the line crosses the y-axis.
02

Plot the y-intercept on the graph

The y-intercept is -2, so place a point on -2 on the y-axis. This point is (0, -2).
03

Use the slope to plot the next point

From the y-intercept, count 3 units up (since the slope is 3) and 1 unit to the right (since the slope is rise over run, or up 3 for right 1). Plot a second point here. Repeat this step to plot a few more points.
04

Draw the graph

Connect all the points drawn in the previous step with a straight line to create the graph of the equation \(y=3x-2\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope
The **slope** is a key concept when working with linear equations. It represents how steep the line is on a graph. In the equation \( y = 3x - 2 \), the slope is the number next to \( x \), which is 3. This means that for every 1 unit increase in \( x \), \( y \) increases by 3 units. So, the slope is essentially the ratio of the rise (the change in \( y \)) over the run (the change in \( x \)). This gives you an idea of the line's direction and steepness:
  • Positive Slope: The line goes upwards from left to right.
  • Negative Slope: The line goes downwards from left to right.
  • Zero Slope: The line is horizontal.
  • Undefined Slope: The line is vertical.
A higher absolute value of the slope means a steeper line. Understanding slope is crucial for graphing and interpreting linear equations.
Y-intercept
The **y-intercept** is another fundamental part of understanding linear equations. In the equation \( y = 3x - 2 \), the y-intercept is -2. This term does not involve \( x \), indicating where the line crosses the y-axis. In practical terms, it tells you the starting value of \( y \) when \( x \) is 0. Here is why it's important:
  • The y-intercept helps to graph the line as it provides a specific point.
  • It shows real-world applications, such as initial conditions before any change occurs.
  • When graphing, the y-intercept provides one of the easiest points to plot.
Understanding the y-intercept helps visualize how linear equations set their baseline on graphs.
Graphing Linear Equations
**Graphing linear equations** involves plotting points on a graph and drawing a line through them. The basic aim is to represent the solution set of the equation visually. Here's how you can do it with an equation like \( y = 3x - 2 \):
  • Start with the y-intercept, which is -2. Plot this on the y-axis at point (0, -2).
  • Use the slope to determine the next points. For a slope of 3, go up 3 units and right 1 unit to find the next point.
  • Repeat the step above to ensure accuracy with more points.
  • Draw a straight line through all the plotted points.
This line is the graph of the equation, representing all solutions to the equation in a coordinate plane.
Coordinate Plane
The **coordinate plane** provides a landscape where linear equations like \( y = 3x - 2 \) can be graphically represented. It consists of a horizontal axis (x-axis) and a vertical axis (y-axis) that intersect at the origin (0,0). Here's how the coordinate plane helps in graphing:
  • The x-axis and y-axis divide the plane into four quadrants, helping in the placement of points.
  • This system allows you to pinpoint exact locations, using pairs \((x, y)\).
  • Linear equations, graphed as lines, show how \( y \) changes with \( x \) across the plane.
The coordinate plane is essential for visualizing how equations transform into graphs and for understanding the spatial relationships between their elements.

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Most popular questions from this chapter

In Exercises 7-12, test for symmetry with respect to \(\theta=\pi / 2\), the polar axis, and the pole. \(r=5+4 \cos \theta\)

\(r=16 \cos 3 \theta\)

\(r=\cos 2 \theta\)

The planets travel in elliptical orbits with the sun at one focus. Assume that the focus is at the pole, the major axis lies on the polar axis, and the length of the major axis is \(2 a\) (see figure). Show that the polar equation of the orbit is \(r=a\left(1-e^{2}\right) /(1-e \cos \theta)\) where \(e\) is the eccentricity.

In Exercises 29-32, use a graphing utility to graph the rotated conic. $$ r=\frac{3}{3+\sin (\theta-\pi / 3)} $$
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